Definition for "Resolvent Operator" in the Quantum Dictionary

- Modern Physics -

Quantum Mechanics Dictionary

A mathematical structure used to represent quantum states and the operations performed on them. It specifies how states are expressed, related, combined, and decomposed into components that support calculation. It does not by itself determine physical predictions without a mapping to observables and measurement rules.

Natural & Formal Sciences Dictionary

For a linear operator A on a Banach space, the resolvent operator at λ ∈ C is R(λ;A) = (A - λ I)^{-1} defined on the resolvent set where A - λ I is invertible and the inverse is bounded. It provides a parameter-dependent bounded inverse outside the spectrum.

- Natural & Formal Sciences -

Mathematics & Logic Dictionary

The family of operator-valued functions R(z) = (A − zI)^{-1} (or (I − zA)^{-1} depending on convention) defined for complex parameters z in the resolvent set where the inverse exists as a bounded operator; used to analyze spectral properties and evolution generated by A.